Imre Vajkovics

Élementa arithmetiae numericae et literalis practicae et theoreticae in usum discentium

proposita. Cassoviae, 1753

Promotore Emer. Vajkovics,

Promotore reverendo patre, Anno MDCCLIII, Junio, Die Prima

Péter Körtesi

Abstract

Imre Vajkovics (Oradea, July 22, 1715 - Kalocsa, November 28, 1798) Doctor of Theology

and Philosophy, Priest of the Society of Jesus, later Great Provost (archbishop)of Kalocsa has

published and has presented his arithmetics in Kosice, 1753. on the 1st of June.

Élementa arithmetiae numericae et literalis practicae et theoreticae in usum discentium

proposita. Cassoviae, 1753.

Promotore Emer. Vajkovics, Promotore reverendo patre, Anno MDCCLIII, Junio, Die Prima.

His presentation and volume edited in Kassa (Cassoviae, Kosice), may be consider as part of

the reform of the Nagyszombati Egyetem, University of Trnava, initiated that year, in order to

introduce natural science subjects in teaching, beside initial Theology and Philosophy

education.

Keywords: Elements of Arithmetics, Educational reform, Nagyszombati Egyetem -

University of Trnava, Kassa - Cassoviae – Kosice

Mathematical subject classification 01A55

Biographical data

Vajkovics Imre SJ (1715. July 25.– Kalocsa, 1798. November 28.): archbishop, great provost

of Kalocsa.

Publications: Elementa arithmeticae... Kassa-Kosice, 1753. – Cels. ac Rev. S. R. I. princeps

Nicolaus e comitibus Csáky de Keresztszegh, ..., laudatione funebris celebratus, ...

Nagyszombat, 1757. – Oratio gratulatoria... Bécs, 1788. – Decas dissertationum ecclesiastico-

politicarum de censoria librorum disciplina, ... Pest, 1791. – Dissertatio de censura librorum

perniciosorum, ... Kalocsa, 1791. – Dissertatio de potestate principum saecularium in

censuram librorum. Uo., 1791. – Quaestio prodroma de voto, ... Uo., 1792. – Censura

religionario-politica libelli, ... H.n., 1792. – Cura superfluorum... Pest, 1792. – De censoria

librorum disciplina libri duo. Kalocsa, 1795. – Iconismus orationis sacrae praetice...

Nagyszombat, 1756; Kalocsa, 1796. – Systema de origine sacrae regni Hung. coronae. 1–2.

köt. Kalocsa, no year 88. [1]

His first publication

Élementa arithmetiae numericae et literalis practicae et theoreticae in usum discentium

proposita. Cassoviae, 1753.

Promotore Emer. Vajkovics, Promotore reverendo patre, Anno MDCCLIII, Junio, Die Prima.

Title pages of the volume

The page below is a courtesy copy got from Dr. Lakatos Andor, head librarian of Archives of

the Kalocsai Főegyházmegyei Levéltár..

Figure 1. Title page 1. of the Vajkovics volume

Figure 2. Title page 2. of the Vajkovics book, delivering information on the place and data of

the presentation of the volume, Aula Universitatis, Anno MDCCLIII. Mense Junio Die Prima

Name of the promoted students

Nomina Promotorum,.

According to that time university tradition the volume which is published by the Jesuit

University of Kassa-Kosice ( Cassoviae), they printed each year a few booklets their

graduates, and the volume usually contained the name of that year promoted students.

The bracelet on the left probably marks the assessment of the students (1-4), see next figure,

actually it could be Magna cum laude, Summa cum laude, Cum laude and Rite.

Figure 3. List of the name of the promoted students (pages 1-2)

Figure 4. List of the name of the promoted students (page 3.)

The content of the volume, Index Sectionum & Capitum is contained at pages 143-144.

Section I

§1. Cap. I. De primo Algorithmo numerico

Notation and use of numbers

§30. Cap. II. Quatuor species literales

Algebraic notations, expressions

§52. Cap. III De Potentiis arithmeticis, &extractione radicum

Power and root definitions

§76. Cap. IV. De Analysi Arithmetica ubi traduntur Regulae generales Analysis

Rules of Arithmetics

Sectio II. De ratione & proportione tum Arithmetica tum Geometrica

§86. Cap. I. Definitiones & Axiomata

Definitions and Axioms

§88. Cap. II De proportione Arithmetica

Arithmetic proportions

§88. Cap. II De proportione arithmetrica

Arithmetric proportions

§103. Cap. III De proportione Geometrica

Geometric proportions

§103. Cap. III De proportione Geometrica

Geometric proportions

§132. Cap. IV Usus Regulae aureae simplices tam directe quam inverse declaratur per

exempla

Use of the simple Golden section, direct or inverse through examples

§143. Cap. V Exempla Regula compositae

Example of the compound Rule

Sectio III. De Fractionibus

About Fractions

§151. Cap. I Definitiones

Definitions

§152. Cap. II. Fractionum 5 species

5 type of Fractions

§174. Cap. III. De fractionibus cum integris

Fractions and Integers

§179. Cap. IV De fractionibus decimalibus

Decimal fractions

Sectio IV.

§190. Applicatio Analysis ad Problemata algebraica

Applications to Algebraic Problems

Cap. I. Applicatio Analysis ad Problemata simplicia @ Determinata

Applications to simple and determined problems

§214. Cap. II. Applicatio Analysis ad Problemata simplicia sed indeterminate.

Applications to simple and undetermined problems

§217. Cap. III. Applicatio Analysis ad aequationes quadraticas

Applications for quadratic equations

Figure 5. Index page 1.

Figure 6. Index page 2.

Mathematical content

We cite here some of the mathematical problems from the volume to offer a self-test for the

reader, how mathematics written in Latin, the universal language of science of the epoch is

still actual.

Figure 7. Adding numbers

Figure 8. Example of adding numbers

Figure 9. Subtraction of numbers

Figure 10. Example for subtraction

Figure 11. Multiplying integers

Figure 12. Example of multiplying integers

Figure 13. Multiplying table

Figure 14. Division of integers

Figure 15. Example for exact integer division

Figure 16. Example for integer division with rest

Figure 17. Algebraic notations

Figure 18. Example for basic algebraic operations, addition, subtraction

Figure 19. Example for basic algebraic operations, multiplication

Figure 20. Example for basic algebraic operations, division

Figure 21. Notations for exponents and radicals

Figure 22. Powers of numbers

Figure 23. Proof of the short algebraic formula of the square of (X+y)

Figure 24. Proof of the short algebraic formula of the square of (X+y+z)

Figure 25. Example of the solution of a word problem, by a linear system of equations

Personal remark

I wish to dedicate, the above Figure 23. as a small present as well for the president of the

János Bolyai Mathematical Society, prof. Pálfy Péter-Pál, to thank him joining us for his

welcome address. The page contains the solution of a word problem about Petrus and Paulus.

Finally, the last problem given in the volume we presented here, is intitled by coincidence

Problema ultimum:

Figure 26. Example of a word problem leading to a quadratic equation

Figure 27. The end of the solution of the above problem

Conclusions

The volume is an example of the usual 1-3 printouts published in the honour of the graduated

students, including the list of names of the graduates. The same year of 1753 we include

below another example, the copy of the title page of a similar volume in Mathematics. Both,

the Arithmetic of Vajkovics and this Mathematics were related to the educational reform in

1753, introduced by Van Swieten, Dutch medical doctor in the court of Maria Theresa. The

reform reduced the three year higher education in to two years, and the Natural Sciences got

more importance than before. The subject of mathematics was moved from the second year of

studies to the first year, and to obtain graduation became more difficult, the number of

graduated people has been drastically reduced.

This system was introduced in 1753 in the Jesuit universities in Nagyszombat -Trnava, Kassa-

Kosice, and Kolozsvár-Cluj as well. The Arithmetics of Vajkovics is due to the consequences

of the educational reform, see another such volume below.

Figure 28. Similar volume on Mathematics, edited in Nagyszombat, Tyrnaviae University, in

the same year 1753

Acknowledgments

I wish to thank Dr. Szögi László for information about the Nagyszombati Egyetem, and Dr.

Lakatos Andor, head librarian of Archives of the Kalocsai Főegyházmegyei Levéltár for

details about the volume of Vajkovics Imre, and especially for the courtesy copy of the title

page missing in the copy I studied.

I wish to thank as well for the president of the János Bolyai Mathgematical Society, prof.

Pálfy Péter-Pál for his welcome speech, dedicating him the Figure 23. the solution of a word

problem about Petrus and Paulus.

References

1. Vajkovics Imre in Magyar Katolikus Lexikon,

[on-line http://lexikon.katolikus.hu/V/Vajkovics.html, retrieved on 18 May 2020.]

http://lexikon.katolikus.hu/V/Vajkovics.html